A291588 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and k >= 0, gcd(a(n), a(n + 2^k)) = 1.

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 14, 27, 25, 16, 31, 37, 12, 35, 41, 22, 43, 39, 20, 47, 49, 32, 33, 53, 26, 59, 61, 15, 67, 71, 28, 73, 45, 34, 79, 77, 38, 65, 83, 46, 89, 21, 40, 97, 91, 44, 51, 95, 58, 101, 103, 18, 55, 107, 52, 109

Offset: 1

Author

Rémy Sigrist, Aug 27 2017

Keywords

• nonn

Comments

For a nonempty subset of the natural numbers, say S, let f_S be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0 and s in S, gcd(a(n), a(n + s)) = 1:

- f_S is well defined (we can always extend the sequence with a new prime number),

- f_S(1) = 1, f_S(2) = 2, f_S(3) = 3,

- all prime numbers appear in f_S, in increasing order,

- if a(k) = p for some prime p, then k <= p and max_{i=1..k} a(i) = p,

- in particular:

S f_S

--------- ---

{ 1 } A000027 (the natural numbers)

{ 2 } A121216

{ 1, 2 } A084937

{ 1, 2, 3 } A103683

{ 1, 2, 3, 4 } A143345

A000027 A008578 (1 alongside the prime numbers)

A000079 a (this sequence)

- see also Links section for the scatterplots of f_S for certain classical S sets,

- likely f_S = f_S' iff S = S'.

The motivation for this sequence is to have a sequence f_S for some infinite subset S of the natural numbers.

Examples

			a(1) = 1 is suitable.
a(2) must be coprime to a(2 - 2^0) = 1.
a(2) = 2 is suitable.
a(3) must be coprime to a(3 - 2^0) = 2, a(3 - 2^1) = 1.
a(3) = 3 is suitable.
a(4) must be coprime to a(4 - 2^0) = 3, a(4 - 2^1) = 2.
a(4) = 5 is suitable.
a(5) must be coprime to a(5 - 2^0) = 5, a(5 - 2^1) = 3, a(5 - 2^2) = 1.
a(5) = 4 is suitable.
a(6) must be coprime to a(6 - 2^0) = 4, a(6 - 2^1) = 5, a(6 - 2^2) = 2.
a(6) = 7 is suitable.
a(7) must be coprime to a(7 - 2^0) = 7, a(7 - 2^1) = 4, a(7 - 2^2) = 3.
a(7) = 11 is suitable.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..20000
• Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000040 (the prime numbers)
• Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000045 (the Fibonacci numbers)
• Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000142 (the factorial numbers)
• Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000244 (the powers of 3)
• Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000312 (A000312(k) = k^k)
• Rémy Sigrist, PARI program for A291588
• Rémy Sigrist, Scatterplot of the first 500000 terms

Crossrefs

Cf. A000027, A000040, A000045, A000079, A000142, A000244, A000312, A008578, A084937, A103683, A121216, A143345.

Programs

PARI

\\ See Links section.